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A New Model for the Origin of the Moon

Thursday, October 18 2012 - 2:14 pm, PDT

Matija Ćuk and Sarah Stewart propose a new model to explain the remarkably similar chemistry of the Earth and Moon. A giant impact onto a fast-spinning Earth ejects material from Earth into orbit, which forms a Moon that is depleted in iron and has a composition similar to Earth's mantle. After the impact, the rapidly rotating Earth is slowed down by a gravitational interaction between the Sun and the Moon called an orbital resonance.

The leading theory for the origin of the Moon is a giant impact with the young Earth. Recently, the hypothesis has been called into question by measurements that find that the Earth and Moon the same isotopic composition (isotopes of an element have slightly different masses). The isotopes of oxygen and titanium, for example, vary widely in the Solar System and are used to ‘fingerprint’ different planets and meteorite groups. The data show that the Earth and Moon are ‘isotopic twins’, but the original giant impact model predicted that most of the Moon was made from the body that struck Earth, which should have had a different isotopic fingerprint.

Therefore, the original giant impact model has a major problem: it can match the mass of the Moon and the rotation rates of the Earth and Moon, but not the chemistry of the Moon. Today, tides between the Earth and Moon slow Earth’s rotation and push the Moon’s orbit further away, but the total angular momentum (see glossary) is conserved. Going back in time, the early Earth had a day of only 5 hours when the Moon formed. With a post-impact spin period of about 5 hours, a giant impact could not loft enough Earth material into orbit to make the Moon match the chemistry of the Earth.

Ćuk and Stewart show that if Earth’s initial angular momentum were higher, corresponding to an Earth day between 2 and 3 hours, a giant impact can eject enough Earth material into orbit to make a Moon with the same isotopic fingerprint. A day of only 2 hours is near the point when the Earth would begin to fly apart from rotational forces. When the Earth is spinning near this rotation limit, it is much easier to launch Earth material into orbit during a giant impact.

Furthermore, Ćuk and Stewart found that the early Earth can have a shorter spin period after the giant impact and then later reach the present spin by transfering angular momentum to the Sun through the 'evection' resonance. The evection resonance is a gravitational interaction between Earth’s orbit around the Sun and the Moon’s orbit around Earth. This new work shows for the first time that it is possible for the early Earth to have had a spin period of only 2 to 3 hours after the giant impact. Now, the giant impact followed by an orbital resonance between the Moon and Sun can explain the chemistry of the Moon and the rotation rates of the Earth and Moon.

Earlier scientists speculated about a fast-spinning early Earth. In 1879, George H. Darwin, a son of Charles Darwin who studied tides, suggested that the Moon formed by fission from the Earth (spinning off material from the Earth), but he did not know how to make the early Earth spin so quickly. Modern studies of planet formation show that Earth grew by a series of giant impacts that made the early Earth spin near its rotational stability limit of about 2 hours. The last giant impact formed a Moon that is a twin of the Earth.

The Giant Impact and Moon-Forming Disk

Calculations of the giant impact event model the collision itself and the generation of a stable disk around the Earth over a period of about 24 hours. The methods used for this stage of Moon formation are not able to model directly the formation of the Moon from the disk, a process which occured over 100's of years as the hot disk cooled. A candidate Moon-forming disk is one that has enough mass and angular momentum to make a satellite the mass of the Moon at the Roche radius (about 3 Earth radii, the distance where a satellite is not broken up by tidal forces). The Moon is 1.2% the mass of the Earth and the disk typically has about twice the mass of the Moon.

Example 1: A Potential Moon-Forming Impact Event

Download higher resolution movies from top right panel. Or click on the image to play movie in a new window.

In the animation, Earth and the smaller planet that hit Earth, named Theia, are represented by many particles with a fixed mass, shown as small balls. The color of the ball indicates the material: iron cores and rocky mantles. Before the impact, Earth's shape is an oblate spheroid because the day is only 2.3 hours long.

In this potential Moon-forming impact, Theia has half the mass of Mars and strikes at 20 km/s. Theia penetrates all the way to the core of the Earth and throws material out, temporarily forming a deep hole in the planet. Theia and part of the Earth are vaporized and expand around the planet. Some material is ejected quickly enough to escape the Earth.

The final disk is massive enough to make the Moon and composed primarily of material from Earth (green balls). The disk has almost no iron as the Theia's iron core merges with Earth's core. This impact scenario agrees with the observed the masses of the Earth and Moon, the low iron content of the Moon, and the similar isotopic composition of the Moon and Earth's mantle. After the impact, the Earth has a day of 2.7 hours.

Example 2: An Unsuccessful Giant Impact

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This animation shows the lower hemisphere of the two colliding planets cut through the equator of the Earth. Each planet is represented by many particles with equal masses. The colors show the iron cores and rocky mantles.

In this example, the growing Earth, with a 3.1 hour spin period, is hit by a Mars-mass body at 9 km/s. Because this impact is slower and more oblique compared to the case shown above, a portion of the projectile shears during the initial contact and sends a spiral arm and fragments into orbit around the Earth. Some large fragments impact the Earth and some remain in orbit. The final disk is massive enough to make a satellite as large as the Moon, but about 60% of the disk originated from the impactor's mantle (yellow balls). Thus, this disk is unlikely to make a Moon with the same isotopic signature as Earth.

Orbital Evolution of the Moon and Angular Momentum Loss

Download higher resolution movies from top right panel. Or click on the image to play movie in a new window.

A top down view of the Moon's orbit around the Earth, with the Sun's direction fixed to the right. The red line shows the orbit of the Moon as it evolves through Earth tides and the evection resonance.

After the giant impact, the Earth-Moon system had more angular momentum than present day. After the Moon forms from the disk, the Moon's orbit begins to expand outward through tidal forces. The Moon is soon caught in the evection resonance, which fixes the closest point in the lunar orbit to be 90 degrees from the Sun. While the Moon is caught in the resonance, typically for several 10,000 years, the Earth is transferring angular momentum to the Moon via tides. But the Moon cannot absorb the angular momentum while it is caught in the resonance. Instead, the Moon passes the angular momentum to the Sun through the resonance. As a result, the Earth's orbit around the Sun expands slightly. After the Moon breaks out of the resonance, it's orbit continues to migrate outward through normal tides with the Earth. Today, the Moon orbits at about 60 Earth radii. While caught in the evection resonance, the Moon would have appeared almost 20 times larger in the sky during its closest approach to Earth.

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Glossary

Angular momentum is an expression of an object's mass and rotational speed. Angular momentum is conserved, which means that the total angular momentum of a group of objects remains the same over time. Angular momentum can be transfered from one object to another.

An orbital resonance is periodic gravitational interaction between two planetary bodies.

The evection resonance occurs when the period of the precession of the lunar orbit equals one year, the period of Earth's orbit around the Sun. In the resonance, gravitational interactions between the Moon and Sun fix the orientation of the Moon's orbit to the Sun (see animation at bottom of page).

More Information

Giant impacts were common at the end of planet formation. Image credit: Don Davis

This work on the origin of the Moon was funded by NASA’s Origins of Solar Systems program and used the Odyssey cluster supported by the Harvard FAS Research Computing Group.